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Bridges may undergo structural vibration responses when exposed to seismic waves. An analysis of structural vibration characteristics is essential for evaluating the safety and stability of a bridge. In this paper, a signal time-frequency feature extraction method (NTFT-ESVD) integrating standard time-frequency transformation, singular value decomposition, and information entropy is proposed to analyze the vibration characteristics of structures under seismic excitation. First, the experiment simulates the response signal of the structure when exposed to seismic waves. The results of the time-frequency analysis indicate a maximum relative error of only 1% in frequency detection, and the maximum relative errors in amplitude and time parameters are 5.9% and 6%, respectively. These simulation results demonstrate the reliability of the NTFT-ESVD method in extracting the time-frequency characteristics of the signal and its suitability for analyzing the seismic response of the structure. Then, a real seismic wave event of the Su-Tong Yangtze River Bridge during the Hengchun earthquake in Taiwan (2006) is analyzed. The results show that the seismic waves only have a short-term impact on the bridge, with the maximum amplitude of the vibration response no greater than 1 cm, and the maximum vibration frequency no greater than 0.2 Hz in the three-dimensional direction, indicating that the earthquake in Hengchun will not have any serious impact on the stability and security of the Su-Tong Yangtze River Bridge. Additionally, the reliability of determining the arrival time of seismic waves by extracting the time-frequency information from structural vibration response signals is validated by comparing it with results from seismic stations (SSE/WHN/QZN) at similar epicenter distances published by the USGS. The results of the case study show that the combination of dynamic GNSS monitoring technology and time-frequency analysis can be used to analyze the impact of seismic waves on the bridge, which is of great help to the manager in assessing structural seismic damage. [ABSTRACT FROM AUTHOR]

An important challenge in the study of complex systems is to identify appropriate effective variables at different times. In this paper, we explain why structures that are persistent with respect to changes in length and time scales are proper effective variables, and illustrate how persistent structures can be identified from the spectra and Fiedler vector of the graph Laplacian at different stages of the topological data analysis (TDA) filtration process for twelve toy models. We then investigated four market crashes, three of which were related to the COVID-19 pandemic. In all four crashes, a persistent gap opens up in the Laplacian spectra when we go from a normal phase to a crash phase. In the crash phase, the persistent structure associated with the gap remains distinguishable up to a characteristic length scale ϵ * where the first non-zero Laplacian eigenvalue changes most rapidly. Before ϵ * , the distribution of components in the Fiedler vector is predominantly bi-modal, and this distribution becomes uni-modal after ϵ *. Our findings hint at the possibility of understanding market crashs in terms of both continuous and discontinuous changes. Beyond the graph Laplacian, we can also employ Hodge Laplacians of higher order for future research. [ABSTRACT FROM AUTHOR]